Counting linear extensions

Brightwell, Graham; Winkler, Peter

doi:10.1007/bf00383444

Cited by 223 publications

(189 citation statements)

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“…Table 9 shows the number of monomials in the resulting polynomials for our random tests. Tables 10,12,11 show the test results of the three methods.…”

Section: 4mentioning

confidence: 99%

“…It is very educational to look first at the case when f is the constant polynomial 1, and the answer is simply a volume. It has been proved that already computing the volume of polytopes of varying dimension is #P-hard [16,12,20,24], and that even approximating the volume is hard [17]. More recently in [28] it was proved that computing the centroid of a polytope is #P-hard.…”

Section: Introductionmentioning

confidence: 99%

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How to integrate a polynomial over a simplex

Berline²,

et al. 2010

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Abstract. This paper settles the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We conclude the article with extensions to other polytopes and discussion of other available methods.

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“…Table 9 shows the number of monomials in the resulting polynomials for our random tests. Tables 10,12,11 show the test results of the three methods.…”

Section: 4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

How to integrate a polynomial over a simplex

Berline²,

et al. 2010

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“…The proof of Lemma 3 constructs a reduction from the #LinearExtension problem, whose #P-hardness has been established in [3].…”

Section: Lemmamentioning

confidence: 99%

“…Fix an instance of the #LinearExtension problem with input S and , and let ϑ be an approximation of Vol(P(S; )) to within an absolute accuracy of ∈ [0, 1/2k!). It is shown in [3] that the number of linear extensions N (S; ) of S under satisfies N (S; ) = k! Vol(P (S; )).…”

Section: #Linearextensionmentioning

confidence: 99%

A comment on “computational complexity of stochastic programming problems”

2015

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Although stochastic programming problems were always believed to be computationally challenging, this perception has only recently received a theoretical justification by the seminal work of Dyer and Stougie (Mathematical Programming A, 106(3): [423][424][425][426][427][428][429][430][431][432] 2006). Amongst others, that paper argues that linear two-stage stochastic programs with fixed recourse are #P-hard even if the random problem data is governed by independent uniform distributions. We show that Dyer and Stougie's proof is not correct, and we offer a correction which establishes the stronger result that even the approximate solution of such problems is #P-hard for a sufficiently high accuracy. We also provide new results which indicate that linear two-stage stochastic programs with random recourse seem even more challenging to solve.

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“…Indeed, counting the number of predecessors of a sequence in the lattice is equivalent to counting the linear extensions of a partial order, which is #P-complete [16]. Furthermore, one needs to take into account the intersection of sets of predecessors, which makes the problem even more difficult.…”

Section: Conclusion and Further Research Directionsmentioning

confidence: 99%